Sparsify and sweep: an efficient preconditioner for the Lippmann-Schwinger equation

TL;DR

This paper introduces a novel preconditioner that combines sparsifying and sweeping techniques to efficiently solve the high-frequency 3D Lippmann-Schwinger equation with near-linear computational cost.

Contribution

The paper presents a new preconditioner with a specially designed PML stencil that transforms dense systems into nearly sparse ones, enabling efficient high-frequency 3D solutions.

Findings

Converges in few iterations with standard GMRES

Iteration count is nearly frequency-independent

Achieves near-linear computational cost in 3D

Abstract

This paper presents an efficient preconditioner for the Lippmann-Schwinger equation that combines the ideas of the sparsifying and the sweeping preconditioners. Following first the idea of the sparsifying preconditioner, this new preconditioner starts by transforming the dense linear system of the Lippmann-Schwinger equation into a nearly sparse system. The key novelty is a newly designed perfectly matched layer (PML) stencil for the boundary degrees of freedoms. The resulting sparse system gives rise to fairly accurate solutions and hence can be viewed as an accurate discretization of the Helmholtz equation. This new PML stencil also paves the way for applying the moving PML sweeping preconditioner to invert the resulting sparse system approximately. When combined with the standard GMRES solver, this new preconditioner for the Lippmann-Schwinger equation takes only a few iterations to converge for both 2D and 3D problems, where the iteration numbers are almost independent of the frequency. To the best of our knowledge, this is the first method that achieves near-linear cost to solve the 3D Lippmann-Schwinger equation in high frequency cases.